Mean atomic distance

Fig. 3. (a) Variation of the unit cell parameters and volume per formula unit with doping, x. (b) Dependence of the mean atomic distances and bond angles with doping, x. Shaded area indicates the x range where the structural transition takes place. 

Table 2. Mean atomic distances and bond angles for the series of ABQb (A = Ndas Sraa ...

A molecular fitting algorithm requires a numerical measure of the difference between two structures when they are positioned in space. The objective of the fitting procedure is to find the relative orientations of the molecules in which this function is minimised. The most common measure of the fit between two structures is the root mean square distance between pairs of atoms, or RMSD ... 

The Rutile Structure.—A large number of compounds MX crystallize with the tetragonal structure of rutile, TiCfe. In this structure the position of the ion X is fixed only by the determination of a variable parameter by means of the intensity of reflection of x-rays from various crystal planes. In accordance with the discussion in a following section, we shall assume the parameter to have the value which causes the distances between X and the three ions M surrounding it to be constant. With this requirement the inter-atomic distance R and the edges a and c of the unit of structure are related by the equation R = (a/4 /2) [2 + (c/o)2]. In this way the inter-atomic distances in Table XII are obtained. In the case of magnesium fluoride the agreement is satisfactory. 

One factor which should be noted for palladium, which also applies to the observation of the transition metals Is that not all crystallites have the same efficiency for diffracting electrons, l.e., as the atomic number decreases, the extinction distance for the crystallite Increases (13). Thus one would anticipate Chat as the mean atomic number decreases, the crystallites will have Co be progressively larger to enable visual observation on a support such as alumina. 

Often, one needs to compare different 3D structures or conformations of a molecule. That is done internally by the 3D stmcture generation program to weed out too similar conformations of fragments. Another aspect is the need of the computational chemist to compare different generated or experimental structures. A well-established measure is the so-called root mean square (RMS) value of all atom-atom distances between two 3D structures. The RMS value needed here is a minimum value achieved by superimposing the two 3D structures optimally. Before calculating the RMS, the sum of interatomic distances is minimized by optimizing the superimposition in 3D. 

One way to solve the problem of unphysically short atomic distances is to project onto the Rpm subspace only those grid points included in a selected strip (gray area), with width of a (cos a + sin a) in the A per subspace. The slope of RPai shown in Fig. 1 is 0.618..., an irrational number related to the golden mean [( /5 + l)/2 = 1.618...]. As a result, the projected ID structure contains two segments (denoted as L and S), and their distribution follows a ID quasiperiodic Fibonacci sequence [2] (c.f. Table 1). From another viewpoint, the ID quasiperiodic structure on the par subspace can be conversely decomposed into periodic components (square lattice) in a (higher) 2D space. The same strip/projection scheme holds for icosahedral QCs, which are truly 3D objects but apparently need a more complex and abstract 6D... 

A technique becomes surface sensitive if the radiation or particles to be detected travel no more than a few atomic distances through the solid. Figure 3.1 shows that the mean free path, A, of electrons in elemental solids depends on the kinetic energy, but is limited to less than 1-2 nm for kinetic energies in the range 15-1000 eV [16]. 

Fig. 5-32. Nonnalized density profile of oxygen atom and hydrogen atom on an interface (111) plane of platinum electrode in aqueous solution p = atomic density Pbuu, = mean atomic density in aqueous solution x= distance normal to the interface. [From Heinzinger, 1993.]...

As mentioned in Sections I.B.2.b and II.A, the dipolar coupling between Li- C may complicate solid state NMR spectra of organolithium compounds and its elimination is often desirable. On the other hand, dipolar coupling constants are related to atomic distances and their determination can yield important structural information. It is therefore of general interest that the REDOR technique, briefly described in Section I.B.2.b, provides a means to determine these parameters. 

To summarize, the existence and role of force in STM is now a well-established scientific fact. At a relatively large absolute distance, for example, 5 A, the force between these two parties is attractive. (By absolute distance we mean the distance between the nucleus of the apex atom of the tip and the top-layer nuclei of the sample surface.) At very short absolute distances, for example, 1.5 A, the force between these two parts is repulsive. Between these two extremes, there is a well-defined position where the net force between the tip and the sample is zero. It is the equilibrium distance. On the absolute distance scale, the equilibrium distance is about 2-2.5 A. Therefore, the tip-sample distance of normal STM operation is 3-7 A on the absolute distance scale. In this range, the attractive atomic force dominates, and the distortion of wavefunctions cannot be disregarded. Therefore, any serious attempt to understand the imaging mechanism of STM should consider the effect of atomic forces and the wavefunction distortions. 

The USR (Ultrafast Shape Recognition) Method. This method was reported by Ballester and Richards (53) for compound database search on the basis of molecular shape similarity. It was reportedly capable of screening billions of compounds for similar shapes on a single computer. The method is based on the notion that the relative position of the atoms in a molecule is completely determined by inter-atomic distances. Instead of using all inter-atomic distances, USR uses a subset of distances, reducing the computational costs. Specifically, the distances between all atoms of a molecule to each of four strategic points are calculated. Each set of distances forms a distribution, and the three moments (mean, variance, and skewness) of the four distributions are calculated. Thus, for each molecule, 12 USR descriptors are calculated. The inverse of the translated and scaled Manhattan distance between two shape descriptors is used to measure the similarity between the two molecules. A value of 1 corresponds to maximum similarity and a value of 0 corresponds to minimum similarity. 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

Random walks on square lattices with two or more dimensions are somewhat more complicated than in one dimension, but not essentially more difficult. One easily finds, for instance, that the mean square distance after r steps is again proportional to r. However, in several dimensions it is also possible to formulate the excluded volume problem, which is the random walk with the additional stipulation that no lattice point can be occupied more than once. This model is used as a simplified description of a polymer each carbon atom can have any position in space, given only the fixed length of the links and the fact that no two carbon atoms can overlap. This problem has been the subject of extensive approximate, numerical, and asymptotic studies. They indicate that the mean square distance between the end points of a polymer of r links is proportional to r6/5 for large r. A fully satisfactory solution of the problem, however, has not been found. The difficulty is that the model is essentially non-Markovian the probability distribution of the position of the next carbon atom depends not only on the previous one or two, but on all previous positions. It can formally be treated as a Markov process by adding an infinity of variables to take the whole history into account, but that does not help in solving the problem. 

Fig. 16. Molecular structure of the core atoms of [(Ph3Sn)3Cr(CO)4] phenyl groups have been omitted for clarity. Selected mean interatomic distances (A) Cr-Sn, 2.71(2) Cr-C, 1.86(2) C-O, 1.15(1).

The result of the solidification of a liquid metal is mostly an unperfect packing, which means a crystal lattice with imperfections. These imperfections have already been discussed in chapter 4. An imperfection is a deviation from the perfect packing. For instance a vacancy is an empty place in the lattice which should be occupied. It is also possible for a particle to end up in a place where it should not be. A foreign metal ion which is bigger than the own ions can upset the move of parts of the lattice in relation to each other. And then there are the faults which we call dislocations and which concern parts of the crystal lattice. For example when it seems as if a crystal is partly cleaved or both parts have shifted over one atom distance in relation to each other. Or the crystal is in fact partly cleaved, both halves move and the crack is filled with a layer of atoms. 

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